A030628 -id:A030628 - cp's OEIS frontend

A292286 a(n) = k if the product of the divisors of n is n^k for some integer k, or -1 if no such k exists. For the ambiguous case, define a(1) = 0.

0, 1, 1, -1, 1, 2, 1, 2, -1, 2, 1, 3, 1, 2, 2, -1, 1, 3, 1, 3, 2, 2, 1, 4, -1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, -1, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, -1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, -1, 2, 4, 1, 3, 2, 4, 1, 6, 1, 2, 3, 3, 2, 4, 1, 5, -1, 2, 1, 6, 2, 2, 2, 4, 1, 6, 2, 3, 2, 2, 2, 6, 1, 3, 3, -1

Offset: 1

Juri-Stepan Gerasimov, Sep 13 2017

sign

If the number of divisors (nd) of n > 1 is odd, then a(n) = -1, else a(n) = nd/2. - Michel Marcus, Sep 14 2017
First occurrence of k beginning with -1 is A293570(r). - Robert G. Wilson v, Oct 10 2017
Records occur for A293570(r): 4, 6, 12, 24, 48, 60, 192, 240, 3072, 12288, 196608, 786432, 12582912, 805306368, etc. - Robert G. Wilson v, Oct 10 2017

			a(10) = 2 because divisors of 10 are 1,2,5,10 with product 100 = 10^2.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from exponents in factorization of n

Numbers n such that the product of divisors of n is n^k: A000040 (k = 1), A007422 (k = 2), A162947 (k = 3), A111398 (k = 4), A030628 (k = 5), A030630 (k = 6).

Cf. A000037, A000290, A056924, A007955, A293570.

Table[Boole[n == 1] + If[OddQ@ #, -1, #/2] &@ DivisorSigma[0, n], {n, 100}] (* Michael De Vlieger, Sep 15 2017 *)

a(n) = if (n==1, 0, my(nd = numdiv(n)); if (nd % 2, -1, nd/2)); \\ Michel Marcus, Sep 14 2017

a(n)=my(k=numdiv(n)); if(k%2, if(n>1, -1, 0), k/2) \\ Charles R Greathouse IV, Sep 19 2017

a(1) = 0, a(A000290(n+1)) = -1, a(A000037(n+1)) = A056924(A000037(n+1)), where A000290 = the squares and A000037 = the nonsquares.

Definition corrected by Charles R Greathouse IV, Sep 13 2017

A065985 Numbers k such that d(k) / 2 is prime, where d(k) = number of divisors of k.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, 38, 39, 44, 45, 46, 48, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 111, 112, 115, 116, 117, 118, 119, 122, 123, 124, 125, 129, 133, 134

Offset: 1

Joseph L. Pe, Dec 10 2001

nonn

Numbers whose sorted prime signature (A118914) is either of the form {2*p-1} or {1, p-1}, where p is a prime. Equivalently, disjoint union of numbers of the form q^(2*p-1) where p and q are primes, and numbers of the form r * q^(p-1), where p, q and r are primes and r != q. - Amiram Eldar, Sep 09 2024

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000005, A118914.
Numbers with exactly 2*p divisors: A030513 (p=2), A030515 (p=3), A030628 \ {1} (p=5), A030632 (p=7), A137485 (p=11), A137489 (p=13), A175744 (p=17), A175747 (p=19).
Subsequences: A006881, A030078, A050997, A054753, A095990, A096156, A138031, A178739, A179665, A189987.

Select[Range[1, 1000], PrimeQ[DivisorSigma[0, # ] / 2] == True &]

n=0; for (m=1, 10^9, f=numdiv(m)/2; if (frac(f)==0 && isprime(f), write("b065985.txt", n++, " ", m); if (n==1000, return))) \\ Harry J. Smith, Nov 05 2009

is(n)=n=numdiv(n)/2; denominator(n)==1 && isprime(n) \\ Charles R Greathouse IV, Oct 15 2015

A137490 Numbers with 27 divisors.

Original entry on oeis.org

900, 1764, 2304, 4356, 4900, 6084, 6400, 10404, 11025, 12100, 12544, 12996, 16900, 19044, 23716, 26244, 27225, 28900, 30276, 30976, 33124, 34596, 36100, 38025, 43264, 49284, 52900, 53361, 56644, 60516, 65025, 66564, 70756, 73984, 74529

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^26 (subset of A089081), p^2*q^2*r^2 (like 900, 1764, 4356, squares of A007304) or p^2*q^8 (like 2304, 6400, subset of the squares of A030628) where p, q and r are distinct primes. - R. J. Mathar, Mar 01 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000005, A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283.

Select[Range[150000],DivisorSigma[0,#]==27&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

isA137490(n) = numdiv(n)==27 \\ Michael B. Porter, Apr 10 2010

A000005(a(n)) = 27.

Sum_{n>=1} 1/a(n) = (P(2)^3 + 2*P(6) - 3*P(2)*P(4))/6 + P(2)*P(8) - P(10) + P(26) = 0.00453941..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

A182680 a(n) = the largest n-digit number with exactly 10 divisors, a(n) = 0 if no such number exists.

Original entry on oeis.org

0, 80, 976, 9904, 99952, 999952, 9999952, 99999824, 999999536, 9999999824, 99999999536, 999999999567, 9999999999963, 99999999999728, 999999999999856, 9999999999998896, 99999999999999824, 999999999999999952, 9999999999999999856, 99999999999999999568

Offset: 1

Jaroslav Krizek, Nov 27 2010

a(n) = the largest n-digit number of the form p^9 or p^4*q (p, q distinct primes), a(n) = 0 if no such number exists.

Cf. A030628, A182679.

Table[k=10^n-1; While[k>10^(n-1) && DivisorSigma[0, k] != 10, k--]; If[k==10^(n-1), k=0]; k, {n, 20}]

A000005(a(n)) = 10.

a(n) >= A182679(n).

Extended by T. D. Noe, Nov 29 2010

A349699 Triangular numbers with exactly 10 divisors.

Original entry on oeis.org

496, 3321, 13203, 195625, 780625, 2883601, 11527201, 107186761, 407879641, 3487920481, 39155632561, 250123560121, 47622568443841, 95853663421561, 322876778328721, 403230060146161, 3034217580863041, 6333850463213521, 13292221046055841, 25335401515201441

Offset: 1

Jon E. Schoenfield, Nov 25 2021

nonn

All terms are of the form p^4 * q, with primes p < q.

a(3) = 13203 = 3^4 * 163 is the only term for which q = 2*p^4 + 1; for all other terms, q is either 2*p^4 - 1 (e.g., a(1) = 496 = 2^4 * 31) or (p^4 + 1)/2 (e.g., a(2) = 3321 = 3^4 * 41).

			Table showing the first 20 terms and their prime factorizations. Because of the different relationships between the prime factors p and q for different terms (see Comments), neither the values of p nor those of q are nondecreasing.
.
   n               a(n) =   p^4 *         q
  --  -------------------------------------
   1                496 =   2^4 *        31
   2               3321 =   3^4 *        41
   3              13203 =   3^4 *       163
   4             195625 =   5^4 *       313
   5             780625 =   5^4 *      1249
   6            2883601 =   7^4 *      1201
   7           11527201 =   7^4 *      4801
   8          107186761 =  11^4 *      7321
   9          407879641 =  13^4 *     14281
  10         3487920481 =  17^4 *     41761
  11        39155632561 =  23^4 *    139921
  12       250123560121 =  29^4 *    353641
  13     47622568443841 =  47^4 *   9759361
  14     95853663421561 =  61^4 *   6922921
  15    322876778328721 =  71^4 *  12705841
  16    403230060146161 =  73^4 *  14199121
  17   3034217580863041 =  79^4 *  77900161
  18   6333850463213521 = 103^4 *  56275441
  19  13292221046055841 = 113^4 *  81523681
  20  25335401515201441 = 103^4 * 225101761

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A000005, A000217, A030628, A178739.

t[n_] := n*(n + 1)/2; Select[t /@ Range[10^5], DivisorSigma[0, #] == 10 &] (* Amiram Eldar, Nov 26 2021 *)

select(x->(numdiv(x)==10), vector(10^5, k, k*(k+1)/2)) \\ Michel Marcus, Nov 26 2021

cp's OEIS Frontend

A292286 a(n) = k if the product of the divisors of n is n^k for some integer k, or -1 if no such k exists. For the ambiguous case, define a(1) = 0.

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A065985 Numbers k such that d(k) / 2 is prime, where d(k) = number of divisors of k.

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A137490 Numbers with 27 divisors.

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A182680 a(n) = the largest n-digit number with exactly 10 divisors, a(n) = 0 if no such number exists.

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A349699 Triangular numbers with exactly 10 divisors.

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